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# Title: Leibniz-Handschriften zur Mathematik LH 35, 6, 27
# Folio 12r, page id 00068072:0023
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19 N Nemtr. 1703
sunt quaedam ita inquirenda, ut multae tentari
debeant viae, Et talia sunt inprimis ea, in quibus compendia quaeruntus
quae an omnino dentur dubitam ante eventum potest. Ut quarentu
numeri divisores Talis videtur inquisitio qreducendamm  infinite-
simalium affectarum ad puras; cum saperranda sunt in determinata
modo tractabili. Sane varia frustra tentavi dudum, sed videmm, nunc
tandem aditus se apernisse Ane ex eo quod
deo constans deo assumi potest per dc ddv: daz Unde in his evitatur
pvarbitrio
ninia depressio; ettell potest res reduci potest vel ad daxidat
per du: dZ vel ad dav: ddz per d dZ: duo.
ut vaglorem ipsius
primum autem hac usus tum via, utl dy: dx obtinerem sine ad,
itoque pro datis y et X, assumsi y: a = sadz: dZ et X: a = sadz. dx
unde fit dy 3dx = dv: d2. sed quia ita bis differen
Quid ergo si sumamus
bis differentiaadum est ante quam perveniatur ad solas dv, dz, ddr
tiandum est calculus fit paulo prolixior.
daz; ideo fit calculus paulo prolixior. Nunc ergo videor
mihi  posse contentus esse
quia in nostra potestate est assumere constantem per
sit ergo X: a ( sddz: aZ et y: a = dv: d2
ddv: _t ddz.
dz ddv - dvddz,: dZ d2
sit jam n
aequatio Taugertialis pata resolvenda dy: dx = XX  + yy, : aa,
s
et dy: dx (5)
erit 1Z tdr = dvddZo: d (= (ad Z qu saaz: dZ + dpdv: d7 d2 = (8,
quam aequod
da
de
d ddv: dd
daz x et fiet de ddv: sd
et lZ) dd v: adZ =
ded  =) dZ qu Sadzue: dz +de + dvdv de.
de
fietque aequatio
quam differentiato
erit dam= (Sad Z int Z) add : dE + d Z qu fdd Z. dz seu da sad. t + (. 4u ) i
de
+ dZdv
et faciamus
e
et fiet ex 10 fiet deo
+(Z. Lato)  dtZ du e seu, qus ddqe d + 2(ddZ. de)
+qudZ Au sddz: de + dd seu
unde extraheado prodibil
compendio du  dudo.due to vocemus di
et diferentiando fit o (13) (dZe 2(5daZ: AZ) dd 3 l: d3
ddz + L, seu qu sddz: dZ +2 sad Z: dz  + dd ip: de z = 0
1
2t
pe
qu sddq d3e + 2 (d ZdZ) fiaz. de + ad 1. 4  0
- 2 Y ( - ddip. dd2) = (17) (per Y7) tedqe di = dvav. da
(ddze. dz = X e dde t et d H
 est autam dd dd (per 1)   d, du (1+ (dv: di). Ergo ad e = de Ad vede) Atet dv. B siu
sddz: d
unde extrahendo
A do: de = d7 ddo - dvadz: dbdo seu = dao +.dZ dv ddx - ddd dd Z : dzdz  et  ddV
2-  dd ip. dezue
et des d(dvidz) = (13) d3 ddv - dodaz ide dv, adeaque per 12 et B ex 59. (7 fiet
qu Sdezidz L
2 - ddv, F 2 dvdav. duebus (- dvdp . d3 d  ddv: ddzu - dvi du - (dv dv: dz Z
fiet 2q (daz. d2) dl f. d2 + da qze qus dav. d + cax 0. unde
-, + ((, - ddv: ddz)
du(per (10) 10) - (3) des - do: d3
seu Adv: ddque = 2 + dvidque 51, 2, 1- dvid2
, -ddvidaz seu (ddz: dx
et differentiando aequationem i
et quSddz: d2 = 2, C. 4 d Lv: ddz - dv.: de - dua v: qed 3 le sit   lv: de + lodv: dide
qusddz: d, +2 sudz: 47 +
2 = ddv: ddque + 2 H1, - dlv: ddZ) C
pro compendio, idve du f dudr: dd vecetur 2d p: deo et fiet X H(1 - dax: daz) Z ddv: ddque - 2 dip: deo, adeoque
1- ddv: d dZ = dd vddo; dazddz - 2 dis ddd: deodu + dv d V: deo dld. et ddv: ddz =
ita res
seu  = ddv: da Z + 2 f (1 - ddv: dd
1+ dbideo 4 + + a   devat B) +5-4 dp. du ( B. 4 dy a p. deoduo)
vdttur reusi ad quadutures rationales